Langlands correspondence and Bezrukavnikov's equivalence
Course given at the University of Sydney, first semester 2019.
A course in two parts:
1) an attempt to explain what the Langlands program is about from an arithmetical point of view;
2) affine Hecke algebras, Deligne-Langlands conjecture and Bezrukavnikov's equivalence.
Extended abstract and bibliography
Notes by Anna Romanova
(Anna's notes also contain a more complete bibliography for part 1 of the course.)
Below are hand-written notes and some handouts for the course, produced as it progressed.
Notes from Gus Lehrer's course last
semester on algebraic number theory.
<!––Links below will work as I upload the appropriate notes from lectures.––>
Lecture 1: Introduction to reciprocity
Number of solutions handout for first lecture.
Lecture 2: Review of algebraic number theory
Degree 5 solutions handout for second lecture.
Lecture 3: Zeta function and L-functions
Excerpt from Mazur-Stein giving Riemann's approximations to the zeta function via more and more roots.
Lecture 4: (Gus' lecture, notes thanks to Bregje Pauwels) Artin L-functions
Lecture 5: (Gus' second lecture, notes thanks to Bregje Pauwels) Brauer's induction theorem
Lecture 6: Overview of the Sato-Tate conjecture
Example sheet of first 5000 primes for two curves.
From slides of a lecture by Ito containing a manuscript of Sato.
Lecture 7: Infinite Galois theory, overview of global class field theory (Artin's point of view).
Lecture 8: Structure of local Galois groups; local class field theory.
Two interesting historical accounts:
K. Conrad: History of class field theory.
K. Miyake: Takagi's Class Field Theory -- From where? and to where?
Lecture 9: Heuristic derivation of local Langlands for GL(2); basic rep theory of p-adic groups.
Lecture 10: Precise statement of local Langlands for GL(2) for p ≠ 2.
Lecture 11: Why is p = 2 special? Spherical representations and Satake isomorphism.
Lecture 12: Deligne-Langlands conjecture, affine Hecke algebras and Kazhdan-Lusztig equivalence, rough statement of Bezrukavnikov's equivalence.